A 3.50 kg box that is several hundred meters above the surface of the earth is suspended from the end of a short vertical rope of negligible mass. A time-dependent upward force is applied to the upper end of the rope, and this results in a tension in the rope of T(t) = (36.0 N/s)t. The box is at rest at t = 0. The only forces on the box are the tension in the rope and gravity. Required:
a. What is the velocity of the box at (i) t = 1.00 s and (ii) t = 3.00 s?
b. What is the maximum distance that the box descends below its initial position?
c. At what value of t does the box return to its initial position?

a. i. -4.65 m/s ii. -13.95 m/s b. 5.89 m c. 2.85 s

Explanation:

a. What is the velocity of the box at (i) t = 1.00 s and (ii) t = 3.00 s?

We write the equation of the forces acting on the mass.

So, T - mg = ma where T = tension in vertical rope = (36.0 N/s)t, m = mass of box = 3.50 kg, g = acceleration due to gravity = 9.8 m/s² and a = acceleration of box = dv/dt where v = velocity of box and t = time.

So, T - mg = ma

T/m - g = a

dv/dt = T/m - g

dv/dt = (36.0 N/s)t/3.50 kg - 9.8 m/s²

dv/dt = (10.3 m/s²)t - 9.8 m/s²

dv = [(10.3 m/s²)t - 9.8 m/s²]dt

Integrating, we have

∫dv = ∫[(10.3 m/s³)t - 9.8 m/s²]dt

∫dv = ∫(10.3 m/s³)tdt - ∫(9.8 m/s²)dt

v = (10.3 m/s³)t²/2 - (9.8 m/s²)t + C

v = (5.15 m/s³)t² - (9.8 m/s²)t + C

when t = 0, v = 0 (since at t = 0, box is at rest)

So,

0 = (5.15 m/s³)(0)² - (9.8 m/s²)(0) + C

0 = 0 + 0 + C

C = 0

So, v = (5.15 m/s³)t² - (9.8 m/s²)t

i. What is the velocity of the box at t = 1.00 s,

v =  (5.15 m/s³)(1.00 s)² - (9.8 m/s²)(1.00 s)

v = 5.15 m/s - 9.8 m/s

v = -4.65 m/s

ii. What is the velocity of the box at t = 3.00 s,

v =  (5.15 m/s³)(3.00 s)² - (9.8 m/s²)(3.00 s)

v = 15.45 m/s - 29.4 m/s

v = -13.95 m/s

b. What is the maximum distance that the box descends below its initial position?

Since v = (5.15 m/s³)t² - (9.8 m/s²)t and dy/dt = v where y = vertical distance moved by mass and t = time, we need to find y.

dy/dt = (5.15 m/s³)t² - (9.8 m/s²)t

dy = [(5.15 m/s³)t² - (9.8 m/s²)t]dt

Integrating, we have

∫dy = ∫[(5.15 m/s³)t² - (9.8 m/s²)t]dt

∫dy = ∫(5.15 m/s³)t²dt - ∫(9.8 m/s²)tdt

∫dy = ∫(5.15 m/s³)t³/3dt - ∫(9.8 m/s²)t²/2dt

y = (1.72 m/s³)t³ - (4.9 m/s²)t² + C'

when t = 0, y = 0.

So,

0 = (1.72 m/s³)(0)³ - (4.9 m/s²)(0)² + C'

0 = 0 + 0 + C'

C' = 0

y = (1.72 m/s³)t³ - (4.9 m/s²)t²

The maximum distance is obtained at the time when v = dy/dt = 0.

So,

dy/dt = (5.15 m/s³)t² - (9.8 m/s²)t = 0

(5.15 m/s³)t² - (9.8 m/s²)t = 0

t[(5.15 m/s³)t - (9.8 m/s²)] = 0

t = 0 or [(5.15 m/s³)t - (9.8 m/s²)] = 0

t = 0 or (5.15 m/s³)t = (9.8 m/s²)

t = 0 or t = (9.8 m/s²)/(5.15 m/s³)

t = 0 or t = 1.9 s

Substituting t = 1.9 s into y, we have

y = (1.72 m/s³)t³ - (4.9 m/s²)t²

y = (1.72 m/s³)(1.9 s)³ - (4.9 m/s²)(1.9 s)²

y = (1.72 m/s³)(6.859 s³) - (4.9 m/s²)(3.61 s²)

y = 11.798 m - 17.689 m

y = -5.891 m

y ≅ - 5.89 m

So, the maximum distance that the box descends below its initial position is 5.89 m

c. At what value of t does the box return to its initial position?

The box returns to its original position when y = 0. So

y = (1.72 m/s³)t³ - (4.9 m/s²)t²

0 = (1.72 m/s³)t³ - (4.9 m/s²)t²

(1.72 m/s³)t³ - (4.9 m/s²)t² = 0

t²[(1.72 m/s³)t - (4.9 m/s²)] = 0

t² = 0 or (1.72 m/s³)t - (4.9 m/s²) = 0

t = √0 or (1.72 m/s³)t = (4.9 m/s²)

t = 0 or t = (4.9 m/s²)/(1.72 m/s³)

t = 0 or t = 2.85 s

So, the box returns to its original position when t = 2.85 s

c.is a substance that requires a small amount of energy per gram to raise its temperature; has strong cohesive forces between molecules.

explanation:

a colorless, transparent, odorless liquid that forms the seas, lakes, rivers, and rain and is the basis of the fluids of living organisms.

ill do it for 100, also this isnt 80 its 40